Question 1

(a)

First, note that the inverse demand curve is given by p(Q)=100-Q/5.
Thus, the firm's revenue is given by

R(Q)=100Q-Q²/5.

Now note that AVC(Q)=10 and fixed costs are 100. Thus, the cost function is given by TC(Q)=10Q+100.
The firm's profit as a function of output is given by

p(Q)=R(Q)-TC(Q)=100Q-Q².5-10Q-100= 90Q-Q²/5-100.

(b)

For an optimal production choice we must have p'(Q)=0, i.e., the derivative of the profit function with respect to Q is zero. Alternatively, we can set MR=MC, where MC is the derivative of R(Q) with respect to Q.
In both cases we get 100-2Q/5=10 which implies 2Q=450. Thus, the firm will produce Q=225 units of output.

Question 2 Recall that at the profit maximizing choice

where e_p is the price elasticity of demand. Thus, we get

Therefore, p=6MC. Because MC=2 the should charge 12 Dollars.

Question 3 A price discriminating monopolist will choose marginal revenue equal to marginal costs at both locations. Then

e revenues at both locations are given by R₁(Q₁)=1,000Q-Q₁² and R₂(Q₂)=500Q₂-2Q₂². Thus, we get 1,000-2Q₁=100 and 500-4Q₂=100. This implies Q₁=450 and Q₂=100. The prices in the markets can be computed by inserting the quantities in the inverse demand curves. Thus, p₁=1000-450=550 an

d p₂=500-2(100)=300. The revenue in both markets is therefore R=450(550)+100(300)=277,500. d'(p)p/d(p). The firm produces a total of 550 units of output. Thus, the production costs are TC(Q)=100(550) +100,000=155,000. The firm's profit is 122,500.

Question 4

• Recall from class the the price elasticity of demand is given by
  e_p=d'(p)p/d(p).

  In this case d(p)=15S^1/2p^-3. The derivative with respect to p is d'(p)=-45S^1/2p^-4. Therefore

  e_p= -45S^1/2p^-4p/(15S^1/2p^-3)= -3

  The price elasticity does not depend on S, and it is constant for all prices.

• At the profit maximizing price
  MC=p(1+1/e_p).

  Therefore MC=(2/3)p, i.e., p=1.5MC. Marginal costs are the derivative of the cost function with respect to output, Q. Thus, MC=0.1. Therefore the profit maximizing price is 15cents.

• Now assume that S=100. The price is 15cents. Inserting these values into the demand function d(p,S)=15S^1/2p^-3 yields d(p)=44,444.

  In general, for arbitrary value of S we get

  Q=15S^1/20.15^-3=4,444.44S^1/2.

• The firm's profits are p=pQ-(10S+0.1Q)=0.15Q-(10S+0.1Q)=0.05Q-10S.
  Inserting Q=4,444.44S^1/2 into this expression yields
  p=0.05(4,444.44S^1/2)-10S= 222.222S^1/2-10S.

• We must find the optimal value for S. To do this we take the derivative of 222.222S^1/2-10S with respect of S and set it equal to 0. Thus, 111.111S^-1/2=10, i.e., S^1/2=11.111, which implies S=123.456.

  Inserting this value and p=0.15 in the demand function yields 49,383.

  Recall that the profit is p= 222.222S^1/2-10S. Thus, the firm's profit is 1,234.5